We have used the fact that a state is separable if and only if its corresponding quantum channel in Eq. (REF ) is a measure-prepare channel [1]}. The latter requires a decomposition of \(v\)  as in Eq. (). Clearly, such a decomposition exists for any commutative subalgebra \(V \subset \mathcal {A}\) . In turn, Thm. REF  shows that such a decomposition exists for all of \(\mathcal {A}\)  if and only if \(\Phi _\rho (\mathcal {A})\)  is a commutative subalgebra.
